Natural densities of Markov transition probabilities (Q1067691)

Let \((P_ t)\), \((P^*\!_ t)\) be two measurable submarkovian semigroups on a measurable space E which are absolutely continuous and in duality with respect to a \(\sigma\)-finite measure \(\mu\). Then it is shown that there exists a unique measurable function \(p: (0,\infty)\times E\times E\to {\bar {\mathbb{R}}}_+\) satisfying \[ (i)\quad P_ tf(x)=\int p(t,x,y)f(y)\mu (dy),\quad P^*\!_ tf(x)=\int p(t,y,x)f(y)\mu (dy)\quad (f\in E_+,\quad x\in E) \] \[ (ii)\quad p(s+t,x,y)=\int p(s,x,z)p(t,z,y)\mu (dz),\quad s,t>0,\quad x,y\in E. \] In particular, for a symmetric semigroup, there exists a unique symmetric density satisfying (i), (ii). A more general result for inhomogeneous transition probabilities is also given.